80 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (d) If C is a convex set in E, the intersection of C with a closed (resp. open) half- space is called a closed (resp. open) cap in C. If a e A is an extremal point, show that the open caps in A which contain a form a fundamental system of neighborhoods of a in A. (Remark that the convex hull of the union of two closed caps in A is weakly compact and does not contain a if neither of the two closed caps contains a\ then use Problem 4(e), and Section 12.3, Problem 6(e).) (e) Suppose that there exists a weakly compact set K c A such that the closed convex hull of K is A. Show that every extremal point of A belongs to K (use (d)). (f) In the Banach space (c0) (Section 5.3, Problem 5) show that the closed ball ||jc|| ^ 1 has no extremal points. 6. Let E, F be two locally convex spaces. Let (pa) be a family of seminorms defining the topology of F, and let @ be a set of bounded subsets of E (Section 12.14, Problem 6). For each continuous linear mapping u of E into F, and each B e @, put pa> B(w) = supj?a(w(jc)). Show that pXtB is a seminorm on the vector space -S?(E; F) of con- xeB tinuous linear mappings of E into F. If the set of homothetic images of the sets of @ covers E, then the topology defined by the seminorms pa, B on JS?(E; F) is Hausdorff. If E and F are normed spaces and if @ consists only of the closed unit ball ||jc|| :g 1 in E, then the corresponding topology on ^(E; F) is that defined by the norm (5.7.1). 7. Let E be an infinite-dimensional separable Hilbert space. Show that, with respect to the topology defined by the norm (5.7.1) on JS?(E; E) (this topology is called the norm topology), the unit ball *ff consisting of the operators U with norm \\U\\ ^1 (the "contractions" of E) is not separable. (Let (en)n^0 be an orthonormal basis of E. For each point z = (£„)«£<> of the Banach space /°° (Section 5.7, Problem 1), let £/ze.S?(E;E) be the mapping defined by Uz-en = £nen for all /zl>0. Show that zt—> Uz is an isometric linear mapping of /°° onto a subspace of «^(E; E), and use the Problem of Section 5,10.) 8. Let E be a separable Hilbert space of infinite dimension. The strong topology on :§?(£; E) is that obtained by the procedure of Problem 6 by taking© to be the set of all finite subsets of E, so that this topology is defined by the seminorms px(U) ~ \\U - x\\ for all x e E. The strong topology is coarser than the norm topology. (a) Show that the strong topology is not metrizable. (Observe that &(E; E) contains vector subs paces which, when given the topology induced by the strong topology, are isomorphic to E endowed with the weak topology.) (b) The unit ball # of ^(E; E), endowed with the strong topology, is separable and metrizable. (Proceed as in (12.5.7), by embedding V in EN.) (c) Show that the mapping (£/, F)h->(7Fis continuous in the strong topology on ^ x &(E; E) and not continuous on J^(E; E^ x #. (d) Show that the mapping Ui-+ U* of <$ into # is not continuous with respect to the strong topology. (Let (<?„)« >o be an orthonormal basis of E, and consider the sequence of operators Uk such that Uk • en = 0 if n < k and Uk • en = en_fc if n ;> k.) (e) Let U(E) c: ^ be the unitary group of E, consisting of all linear isometrics of E onto E. Show that the topology induced on U(E) by the strong topology is com- patible with the group structure. Give an example in this group of a right Cauchy sequence which is not a left Cauchy sequence. Show that U(E) is closed in #. 9. Let E be a separable Hilbert space of infinite dimension. The weak topology on ; E) is that obtained by the procedure of Problem 6 by taking© to be the set of set consisting of a single point is closed,